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For nothing

Let Σ =(,,)\Sigma_\emptyset =(\emptyset,\emptyset,\emptyset) be the empty signature. Then a Σ \Sigma_\emptyset-structure in a Grothendieck topos \mathcal{E} is simply a triple of empty assignments but there is just one of these and it is trivially a model for the empty theory 𝕋 Σ =\mathbb{T}_{\Sigma_\emptyset}=\emptyset over the empty signature since there are no sequents to satisfy and the only 𝕋 Σ \mathbb{T}_{\Sigma_\emptyset}-homomorphism is the empty collection of maps from \mathcal{E} whence Mod 𝕋 Σ ()=1Mod_{\mathbb{T}_{\Sigma_\emptyset}}(\mathcal{E})=\mathbf{1}\!; in other words, the classifying topos Set[𝕋 Σ ]Set[\mathbb{T}_{\Sigma_\emptyset}] is the terminal Grothendieck topos SetSet.

But SetSet has no non-trivial subtoposes which implies that relative to the empty signature the empty theory is complete: either a sequent σ\sigma follows from 𝕋 Σ \mathbb{T}_{\Sigma_\emptyset} or {σ}\{\sigma\} is inconsistent, in other words, the only toposes classifying theories over the empty signature are SetSet and the inconsistent topos 1\mathbf{1}.

Relative to Σ {\Sigma_\emptyset} the models of the theories classified by SetSet and 1\mathbf{1} take on the somewhat ghostlike appearance as empty assigments but enlargening the signature gives them more concrete content: e.g. admitting a sort symbol one sees that 1\mathbf{1} classifies zero objects which of course only degenerate toposes admit and that SetSet classifies initial objects. Whereas the contradictory theory {}\{\top\vdash\bot\} robustly axiomatizes the theories classified by 1\mathbf{1} in arbitrary signatures, in the case of SetSet the theories vary with and within the signature1 e.g. in order to axiomatize initial objects one has to add the sequent x\top\vdash_x\bot to the theory of objects i.e. the empty theory relative to the signature with exactly one sort symbol; more on this in the next section and the entries linked to there.


  1. This difference in behavior is a reflection of the behavior as subtoposes: whereas 1\mathbf{1} is always the bottom of the lattice, SetSet is only constrained to appear as atom; in particular it might appear more than once (cf. the example at hypergraph).

Last revised on May 26, 2020 at 08:46:16. See the history of this page for a list of all contributions to it.